Nonlinear Projections - Transformations and Projections in Computer Graphics - page 166

of line 2 is a half-circle (not shown) that becomes a straight line when the cylinder is
unrolled. This shows how horizontal straight lines are projected by curved perspective
into either horizontal segments or curves. The three segments are projected into the
cylinder in the region bounded by the W and E directions. Two segments become curves
(whose curvature depends on the height of the projected segment), and the central one
remains straight. Vertical lines are always projected into vertical straight segments.
Figure 4.23 is an extension of Figure 4.22e. It illustrates the 360 ◦ cylindrical projec-
tion of horizontal straight segments in four directions. Part (a) of the figure shows four
segments and their directions. Part (b) shows how each segment becomes a curve on
the unrolled cylinder. Segment 1, to the north, is projected into a curve between W and
E (several curves are shown, which are the projections of segments at various heights).
Segment 4, to the south, is projected from E to W through S, so it is displayed in two
halves. Segment 2, to the west, is projected from S through W to N, and segment 3 is
projected from N through E to S. Some straight vertical segments are also shown. Such
a grid corresponds to the continuous four-point perspective of Section 4.8.

S W N E S

W

N

S

E

4 2 1 3 4

2

4

3

1

cut here

(a) (b)

Figure 4.23: (a) Four segments. (b) Cylindrical Projections of Horizontal Segments.

Such a grid is handy when we want to compute or paint the cylindrical projection
of a three-dimensional scene on a rectangular canvas. This can be done either manually
or by special software. Any point in the space around the cylinder of Figure 4.23a is
projected onto the surface of the cylinder by moving it to the surface along the segment
that connects it to the center of the cylinder. Once a point is on the surface of the
cylinder, it is easy to tell where it should go on the grid of Figure 4.23b.

Art, like morality, consists in drawing the line somewhere.
—G. K. Chesterton
A great artist is always before his time or behind it.
—George Moore

Figure 4.24 (courtesy of Dick Termes) is an example of such a drawing. It depicts
a familiar scene, so there is no need to include the original three-dimensional image or
any hints. The reader should especially note how the vertical lines are straight and how